A109189 -id:A109189 - cp's OEIS frontend

A089627 T(n,k) = binomial(n,2k)binomial(2*k,k) for 0 <= k <= n, triangle read by rows.

1, 1, 0, 1, 2, 0, 1, 6, 0, 0, 1, 12, 6, 0, 0, 1, 20, 30, 0, 0, 0, 1, 30, 90, 20, 0, 0, 0, 1, 42, 210, 140, 0, 0, 0, 0, 1, 56, 420, 560, 70, 0, 0, 0, 0, 1, 72, 756, 1680, 630, 0, 0, 0, 0, 0, 1, 90, 1260, 4200, 3150, 252, 0, 0, 0, 0, 0, 1, 110, 1980, 9240, 11550, 2772, 0, 0, 0, 0, 0, 0, 1, 132, 2970, 18480, 34650, 16632, 924, 0, 0, 0, 0, 0, 0, 1, 156, 4290, 34320, 90090, 72072, 12012, 0, 0, 0, 0, 0, 0, 0, 1, 182, 6006, 60060, 210210, 252252, 84084, 3432, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 31 2003

The rows of this triangle are the gamma vectors of the n-dimensional type B associahedra (Postnikov et al., p.38 ). Cf. A055151 and A101280. - Peter Bala, Oct 28 2008
T(n,k) is the number of Grand Motzkin paths of length n having exactly k upsteps (1,1). Cf. A109189, A055151. - Geoffrey Critzer, Feb 05 2014
The result Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)*x^k = (sqrt(1 - 4*x))^n* P(n,1/sqrt(1 - 4*x)) expressing the row polynomials of this triangle in terms of the Legendre polynomials P(n,x) is due to Catalan. See Laden, equation 7.10, p. 56. - Peter Bala, Mar 18 2018

			Triangle begins:
  1
  1,   0
  1,   2,    0
  1,   6,    0,     0
  1,  12,    6,     0,     0
  1,  20,   30,     0,     0,     0
  1,  30,   90,    20,     0,     0,   0
  1,  42,  210,   140,     0,     0,   0, 0
  1,  56,  420,   560,    70,     0,   0, 0, 0
  1,  72,  756,  1680,   630,     0,   0, 0, 0, 0
  1,  90, 1260,  4200,  3150,   252,   0, 0, 0, 0, 0
  1, 110, 1980,  9240, 11550,  2772,   0, 0, 0, 0, 0, 0
  1, 132, 2970, 18480, 34650, 16632, 924, 0, 0, 0, 0, 0, 0
Relocating the zeros to be evenly distributed and interpreting the triangle as the coefficients of polynomials
                     1
                     1
                 1 + 2 q^2
                 1 + 6 q^2
            1 + 12 q^2 +  6 q^4
            1 + 20 q^2 + 30 q^4
       1 + 30 q^2 +  90 q^4 +  20 q^6
       1 + 42 q^2 + 210 q^4 + 140 q^6
  1 + 56 q^2 + 420 q^4 + 560 q^6 + 70 q^8
then the substitution q^k -> 1/(floor(k/2)+1) gives the Motzkin numbers A001006.
- _Peter Luschny_, Aug 29 2011

Alois P. Heinz, Rows n = 0..140, flattened
Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), #14.9.1.
Hyman N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938.
Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013.
A. Postnikov, V. Reiner, and L. Williams, Faces of generalized permutohedra, arXiv:math/0609184 [math.CO], 2006-2007.

Row sums A002426. Antidiagonal sums A098479.

Cf. A055151, A063007, A109187, A101280.

for i from 0 to 12 do seq(binomial(i, j)*binomial(i-j, j), j=0..i) od; # Zerinvary Lajos, Jun 07 2006
# Alternatively:
R := (n, x) -> simplify(hypergeom([1/2 - n/2, -n/2], [1], 4*x)):
Trow := n -> seq(coeff(R(n,x), x, j), j=0..n):
seq(print(Trow(n)), n=0..9); # Peter Luschny, Mar 18 2018

nn=15;mxy=(1-x-(1-2x+x^2-4x^2y)^(1/2))/(2x^2 y);Map[Select[#,#>0&]&, CoefficientList[Series[1/(1-x-2y x^2mxy),{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Feb 05 2014 *)

T(n,k) = binomial(n,2*k)*binomial(2*k,k);
concat(vector(15, n, vector(n, k, T(n-1, k-1)))) \\ Gheorghe Coserea, Sep 01 2018

T(n,k) = n!/((n-2*k)!*k!*k!).
E.g.f.: exp(x)*BesselI(0, 2*x*sqrt(y)). - Vladeta Jovovic, Apr 07 2005
O.g.f.: ( 1 - x - sqrt(1 - 2*x + x^2 - 4*x^2*y))/(2*x^2*y). - Geoffrey Critzer, Feb 05 2014
R(n, x) = hypergeom([1/2 - n/2, -n/2], [1], 4*x) are the row polynomials. - Peter Luschny, Mar 18 2018
From Peter Bala, Jun 23 2023: (Start)
T(n,k) = Sum_{i = 0..k} (-1)^i*binomial(n, i)*binomial(n-i, k-i)^2. Cf. A063007(n,k) = Sum_{i = 0..k} binomial(n, i)^2*binomial(n-i, k-i).
T(n,k) = A063007(n-k,k); that is, the diagonals of this table are the rows of A063007. (End)

A109190 Number of (1,0)-steps at level zero in all Grand Motzkin paths of length n.

Original entry on oeis.org

1, 0, 2, 2, 8, 16, 46, 114, 310, 822, 2238, 6094, 16764, 46308, 128650, 358862, 1005056, 2824416, 7962122, 22508350, 63792424, 181219680, 515905018, 1471593638, 4205280902, 12037415526, 34510499066, 99083855234, 284870069780

Offset: 0

Emeric Deutsch, Jun 21 2005

nonn

A Grand Motzkin path of length n is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) with steps u=(1,1), d=(1,-1) and h=(1,0).
Column 0 of A109189.
The substitution x->x/(1+x+x^2) in the g.f. (this might be called an inverse Motzkin transform), yields the g.f. of (-1)^n*A006355(n). - R. J. Mathar, Nov 10 2008
Apparently also the number of grand Motzkin paths of length n that avoid flat steps at level 0. - David Scambler, Jul 04 2013
Motzkin contexts such that along the path from the root to the hole there are only binary nodes. - Pierre Lescanne, Nov 11 2015

			a(3) = 2 because we have uhd and dhu.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul Barry, Moment sequences, transformations, and Spidernet graphs, arXiv:2307.00098 [math.CO], 2023.
Taras Goy and Mark Shattuck, Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.

Cf. A001006, A006355, A109189.

g:=(sqrt(1-2*z-3*z^2)-z)/(1-2*z-4*z^2): gser:=series(g,z=0,33): 1,seq(coeff(gser,z^n),n=1..30);

CoefficientList[Series[(Sqrt[1-2*x-3*x^2]-x)/(1-2*x-4*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 03 2014 *)

x='x+O('x^55); Vec((sqrt(1-2*x-3*x^2)-x)/(1-2*x-4*x^2)) \\ Altug Alkan, Nov 11 2015

G.f.: (sqrt(1-2*z-3*z^2)-z)/(1-2*z-4*z^2).
G.f.: 1/(1-2x^2*M(x)), M(x) the g.f. of the Motzkin numbers A001006. - Paul Barry, Mar 02 2010
D-finite with recurrence n*a(n) +(3-4*n)*a(n-1) +3*(1-n)*a(n-2) +2*(7*n-15)*a(n-3) +12*(n-3)*a(n-4) = 0. - R. J. Mathar, Nov 09 2012
a(n) ~ 3^(n+3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 03 2014

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A089627 T(n,k) = binomial(n,2k)binomial(2*k,k) for 0 <= k <= n, triangle read by rows.

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A109190 Number of (1,0)-steps at level zero in all Grand Motzkin paths of length n.

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cp's OEIS Frontend

A089627 T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.

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Maple

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A109190 Number of (1,0)-steps at level zero in all Grand Motzkin paths of length n.

Views

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Comments

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Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A089627 T(n,k) = binomial(n,2k)binomial(2*k,k) for 0 <= k <= n, triangle read by rows.